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Random Graphs
and
The Parity Quantifier
Phokion G. Kolaitis
UC Santa Cruz
&
IBM Research-Almaden
Swastik Kopparty
MIT
&
Institute for Advanced Study
What is finite model theory?
It is the study of logics on classes of finite structures.
Logics:
First-order logic FO and various extensions of FO:
 Fragments of second-order logic SO.
 Logics with fixed-point operators.
 Finite-variable infinitary logics.
 Logics with generalized quantifiers.
2
Main Themes in Finite Model Theory




Classical model theory in the finite:
Do the classical results of model theory hold in the finite?
Expressive power of logics in the finite:
What can and what cannot be expressed in various logics on
classes of finite structures.
Descriptive complexity:
computational complexity vs. uniform definability
(logic-based characterizations of complexity classes).
Logic and asymptotic probabilities on finite structures
0-1 laws and convergence laws.
3
Logic and Asymptotic Probabilities



Notation:
 Q:
Property (Boolean query) on the class F of all finite structures
 Fn:
Class of finite structures with n in their universe
  n:
Probability measure on Fn, n ¸ 1
 n(Q) =
Probability of Q on Fn with respect to n, n ¸ 1.
Definition: Asymptotic probability of property Q
(Q) = lim n! 1(Q) (provided the limit exists)
Examples: For the uniform measure  on finite graphs G:

(G contains a triangle) = 1.

(G is connected) = 1.

(G is 3-colorable) = 0.

(G is Hamiltonian) = 1.
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0-1 Laws and Convergence Laws
Question: Is there a connection between the definability of a
property Q in some logic L and its asymptotic probability?
Definition: Let L be a logic
 The 0-1 law holds for L w.r.t. to a measure n, n¸ 1, if
() = 0 or () = 1,
for every L-sentence .

The convergence law holds for L w.r.t. to a measure n, n¸ 1,
if () exists, for every L-sentence .
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0-1 Law for First-Order Logic
Theorem: Glebskii et al. – 1969, Fagin – 1972
The 0-1 law holds for FO w.r.t. to the uniform measure on the class of
all finite graphs.
Proof Techniques:
 Glebskii et al.
Quantifier Elimination + Counting
 Fagin
Transfer Theorem:
There is a unique countable graph R such that for every
FO-sentence , we have that
() = 1 if and only if R ² .
Note:
 R is Rado’s graph: Unique countable, homogeneous, universal
graph; it is characterized by a set of first-order extension axioms.
 Each extension axiom has asymptotic probability equal to 1.
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FO Truth vs. FO Almost Sure Truth
Everywhere true (valid)

Somewhere true &
Somewhere false
Everywhere false (contradiction)
Almost surely true

Almost surely false
First-Order Truth
Testing if a FO-sentence is
true on all finite graphs is an
undecidable problem
(Trakhtenbrot - 1950)
Almost Sure First-Order Truth
Testing if a FO-sentence is
almost surely true on all finite
graphs is a decidable problem; in
fact, it is PSPACE-complete
(Grandjean - 1985).
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Three Directions of Research on 0-1 Laws



0-1 laws for extensions of FO w.r.t. the uniform measure.
0-1 laws for FO on restricted classes of finite structures
 Partial Orders, Triangle-Free Graphs, …
0-1 laws on graphs under variable probability measures.
 G(n,p) with p ≠ ½ (e.g., p(n) = n-(1/e))
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0-1 Laws for Extensions of First-Order Logic
Many generalizations of the original 0-1 law, including:
 Blass, Gurevich, Kozen – 1985
0-1 Law for Least Fixed-Point Logic (LFP)
 Captures Connectivity, Acyclicity, 2-Colorability, …


K … and Vardi – 1990
0-1 Law for Finite-Variable Infinitary Logics Lk1!, k¸ 2
 Proper extension of LFP
K… and Vardi – 1987, 1988
0-1 Laws for fragments of Existential Second-Order Logic
 Capture 3-Colorability, 3-Satisfiability, …
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Logics with Generalized Quantifiers


Dawar and Grädel – 1995
 0-1 Law for FO[Rig], i.e., FO augmented with the rigidity
quantifier.
 Sufficient condition for the 0-1 Law to hold for FO[Q],
where Q is a collection of generalized quantifiers.
Kaila – 2001, 2003
 Sufficient condition for the 0-1 Law to hold for Lk1! [Q],
k¸ 2, where is a collection of simple numerical quantifiers.
 Convergence Law for Lk1! [Q], k¸ 2, where is a collection
of certain special quantifiers on very sparse random finite
structures.
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A Barrier to 0-1 Laws
All generalizations of the original 0-1 law are obstructed by
THE PARITY PROBLEM
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The Parity Problem
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
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Consider the property
Parity = “there is an odd number of vertices”
For n odd, n(Parity) = 1
For n even, n(Parity) = 0
Hence, (Parity) does not exist.
Thus, if a logic L can express Parity, then even the
convergence law fails for L.
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First-Order Logic + The Parity Quantifier
Goal of this work:


Turn the parity barrier into a feature.
Investigate the asymptotic probabilities of properties of finite
graphs expressible in FO[©], that is, in
first-order logic augmented with the parity quantifier ©.
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FO[©]: FO + The Parity Quantifier ©

Syntax of FO[©]: If (v) is a formula, then so is © v (v).

Semantics of © v (v):


“the number of v’s for which (v) is true is odd”
Examples of FO[©]-sentences on finite graphs:
 © v 9 w E(v, w)
The number of vertices of positive degree is odd.
: 9 v © w E(v, w)




There is no vertex of odd degree, i.e.,
The graph is Eulerian.
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Vectorized FO[©]


Syntax: If (v1,…,vt) is a formula, then so is
©(v1,…,vt) (v1,…,vt)
Semantics of ©(v1,…,vt) (v1,…,vt):


“there is an odd number of tuples (v1,…,vt) for which
(v1,…,vt) is true”
Fact:
©(v1,…,vt) (v1,…,vt)
iff
© v1 © v2  © vt (v1,…,vt).
 Thus, FO[©] is powerful enough to express its vectorized
version.
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The Uniform Measure on Finite Graphs
Let Gn be the collection of all finite graphs with n vertices

The uniform measure on Gn:
 If G 2 Gn, then prn(G) = 1/ 2n(n-1)/2
 If Q is a property of graphs, then
prn(Q) = fraction of graphs in Gn that satisfy Q.
An equivalent formulation
 The G(n, 1/2)-model:
 Random graph with n vertices
 Each edge appears with probability ½ and independently of
all other edges
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FO[©] and Asymptotic Probabilities
Question: Let  be a FO[©]-sentence.
What can we say about the asymptotic behavior of
the sequence
prn(), n ¸ 1 ?
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Asymptotic Probabilities of FO[©]-Sentences
Fact: The 0-1 Law fails for FO[©]
Reason 1 (a blatant reason):
Let  be the FO[©]-sentence © v (v = v)
Then

pr2n() = 0
pr2n+1() = 1.
Hence,


limn ! 1 prn () does not exist.
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Asymptotic Probabilities of FO[©]-Sentences
Reason 2 (a more subtle reason):
v5


v1
Let  be the FO-sentence
© v1, v2, …, v6
v6
Fact (intuitive, but needs proof):
limn ! 1 prn() = 1/2
v2
v3
v4
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Modular Convergence Law for FO[©]
Main Theorem: For every FO[©]-sentece , there exist two
effectively computable rational numbers a0, a1 such that

limn ! 1 pr2n() = a0

limn ! 1 pr2n+1() = a1.
Moreover, a0, a1 are of the form s/2t, where s and t are
positive integers.
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In Contrast


Hella, K …, Luosto - 1996
LFP[©] is almost-everywhere-equivalent to PTIME.
Hence, the modular convergence law fails for LFP[©].
Kaufmann and Shelah - 1985
For every rational number r with 0 < r < 1, there is
a sentence  of monadic second-order logic such that
limn ! 1 prn () = r.
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Modular Convergence Law
Main Theorem: For every FO[©]-sentece , there exist two
effectively computable rational numbers a0, a1 of the form s/2t
such that
 limn ! 1 pr2n() = a0
 limn ! 1 pr2n+1() = a1.
Proof Ingredients:
 Elimination of quantifiers.
 Counting results obtained via algebraic methods used in the
study of pseudorandomness in computational complexity.
 Functions that are uncorrelated with low-degree
multivariate polynomials over finite fields.
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Counting Results – Warm-up
Notation: Let H be a fixed connected graph.
 #H(G) = the number of “copies” of H as a subgraph of G
= |Inj.Hom(H,G)| / |Aut(H)|.
Basic Question:
What is pr(#H(G) is odd), for a random graph G?
Lemma: If H is a fixed connected graph, then for all large n,
prn(#H(G) is odd) = 1/2 + 1/2n .
Proof uses results of Babai, Nisan, Szegedy – 1989.
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Counting Results – Subgraph Frequencies
Definition: Let m be a positive integer and let
H1,…,Ht be an enumeration of all distinct connected graphs that
have at most m vertices.

The m-subgraph frequency vector of a graph G is the vector
freq(m,G) = (#H1(G) mod2, …, #Ht(G) mod2)
Theorem A: For every m, the distribution of freq(m,G) in
G(n,1/2) is 1/2n-close to the uniform distribution over {0,1}t,
except for #K1 = n mod2, where K1 is
.
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Quantifier Elimination
Theorem B: The asymptotic probabilities of FO[©]-sentences
are “determined” by subgraph frequency vectors.
More precisely:
For every FO[©]-sentence , there are a positive integer m
and a function g: {0,1}t ! {0,1} such that for all large n,
prn(G ² 
,
g(freq(m,G))=1) = 1-1/2n.
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Quantifier Elimination
Theorem B: The asymptotic probabilities of FO[©]-sentences
are “determined” by subgraph frequency vectors.
Proof: By quantifier elimination.
However, one needs to prove a stronger result about formulas
with free variables (“induction loading device”).
Roughly speaking, the stronger result asserts that:
The asymptotic probability of every FO[©]-formula (w1, …, wk)
is determined by:
 Subgraph induced by w1, …, wk.
 Subgraph frequency vectors of graphs anchored at w1, …, wk.
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Quantifier Elimination
Notation:
 typeG (w1, …, wk) = Subgraph of G induced by w1,…,wk
 Types(k) = Set of all k-types
 freq(m,G,w1, …, wk) = Subgraph frequencies of graphs
anchored at w1, …, wk
w1
w2
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Quantifier Elimination
Theorem B’: For every FO[©]-formula (w1, …, wk), there are
a positive integer m and a function h: Types(k) x {0,1}t ! {0,1}
such that for all large n,
prn(8 w (G ² (w)
,
h(typeG(w), freq(m,G w))=1)) = 1-1/2n.
Note:
 k = 0 is Theorem B.

(w1, …, wk) quantifier-free is trivial:
determined by type.
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Modular Convergence Law for FO[©]
Theorem A: For every m, the distribution of freq(m,G) in
G(n,1/2) is 1/2n-close to the uniform distribution over {0,1}t,
except for #K1 = n mod2, where K1 is .
Theorem B: For every FO[©]-sentence , there are a positive
integer m and a function g: {0,1}t ! {0,1} such that
for all large n, prn(G ²  , g(freq(m,G))=1) = 1-1/2n.
Main Theorem: For every FO[©]-sentence , there exist
effectively computable rational numbers a0, a1 of the form s/2t
such that


limn ! 1 pr2n() = a0
limn ! 1 pr2n+1() = a1.
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Modular Convergence Law for FO[Modq]
Theorem: Let q be a prime number.
For every FO[Modq]-sentece , there exist effectively
computable rational numbers a0, a1, …,aq-1 of the form s/qt
such that for every i with 0 · i · q-1,
lim n ´ i mod q, n ! 1 prn() = ai .
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Open Problems



What is the complexity of computing the limiting probabilities
of FO[©]-sentences?
 PSPACE-hard problem;
…
 In Time(22 ).
Is there a modular convergence law for FO[Mod6]?
More broadly,
 Understand FO[Mod6] on random graphs.
 May help understanding AC0[Mod6] better.
Modular Convergence Laws for FO[©] on G(n, n-a)?
31